In a bowl, mix shredded chicken with 1 cup of Monterey Jack cheese. Fill each tortilla with the mixture, roll tightly, and place seam-side down in a greased 9x13-inch baking dish.
In a skillet, melt butter over medium heat. Stir in flour and cook for 1 minute to form a roux. Slowly whisk in chicken broth until smooth and slightly thickened.
Remove from heat, stir in sour cream and diced green chilies until combined.
Pour the sauce evenly over the tortillas. Sprinkle with the remaining cheese.
Preheat the oven to 350°F. Bake uncovered for 20–25 minutes or until bubbly and golden.
Let cool slightly before serving. Enjoy with your favorite Tex-Mex sides.